TauLib · API Book V

TauLib.BookV.Prologue.HermeticPrinciple

TauLib.BookV.Prologue.HermeticPrinciple

The hermetic principle: the crossed tensor product τ³ = τ¹ ×_f T² is NOT a direct product. Fiber completeness and the temporal complement recap for Book V’s opening chapter.

Registry Cross-References

  • [V.R04] Fiber Completeness — fiber_completeness_count

  • [V.T06] Fiber Sector Coverage — fiber_covers_nongrav

  • [V.R05] Temporal Complement Recap — temporal_complement_recap

Mathematical Content

Fiber Completeness [V.R04]

The crossed tensor product τ³ = τ¹ ×_f T² is NOT a direct product: the fibration map f encodes all inter-sector couplings. Nevertheless, the fiber T² and base τ¹ together exhaust all 5 sectors.

The 3 fiber sectors {B (EM), C (Strong), Omega (Higgs)} account for all non-gravitational, non-temporal physics. The 2 base sectors {D (Gravity), A (Weak)} account for all temporal physics.

Fiber Coverage [V.T06]

The boundary holonomy algebra restricted to the 3 fiber sectors covers all non-gravitational physics: EM, Strong, and Higgs/mass crossing.

Temporal Complement [V.R05]

The Temporal Complement Theorem κ(A;1) + κ(D;1) = 1 (proven in Book IV) means the base sectors fully account for the temporal coupling budget. No coupling “leaks” between base and fiber — the partition is hermetic.

Ground Truth Sources

  • Book V Chapter 1 (2nd Edition): The Self-Describing Universe

  • Book IV Chapter 6: Five Sectors

Tau.BookV.Prologue.FiberCompleteness

source structure Tau.BookV.Prologue.FiberCompleteness :Type

[V.R04] Fiber completeness: the 3 fiber sectors (B, C, Omega) that live on T² in the τ³ = τ¹ ×_f T² fibration.

The crossed tensor product is NOT a direct product — the fibration map f encodes all inter-sector couplings. But the partition into base (temporal) and fiber (spatial) sectors is exact: 2 + 3 = 5.

Fiber sectors: B (EM, γ), C (Strong, η), Omega (Higgs, ω).

  • fiber_sectors : List BookIII.Sectors.Sector The three fiber sectors.

  • fiber_count : self.fiber_sectors.length = 3 Exactly 3 fiber sectors.

  • base_sectors : List BookIII.Sectors.Sector The two base sectors.

  • base_count : self.base_sectors.length = 2 Exactly 2 base sectors.

  • total : self.fiber_sectors.length + self.base_sectors.length = 5 Total = 5.

Instances For

Tau.BookV.Prologue.instReprFiberCompleteness.repr

source def Tau.BookV.Prologue.instReprFiberCompleteness.repr :FiberCompleteness → ℕ → Std.Format

Equations

  • One or more equations did not get rendered due to their size. Instances For

Tau.BookV.Prologue.instReprFiberCompleteness

source instance Tau.BookV.Prologue.instReprFiberCompleteness :Repr FiberCompleteness

Equations

  • Tau.BookV.Prologue.instReprFiberCompleteness = { reprPrec := Tau.BookV.Prologue.instReprFiberCompleteness.repr }

Tau.BookV.Prologue.canonical_fiber_completeness

source def Tau.BookV.Prologue.canonical_fiber_completeness :FiberCompleteness

The canonical fiber completeness instance. Equations

  • One or more equations did not get rendered due to their size. Instances For

Tau.BookV.Prologue.fiber_completeness_count

source theorem Tau.BookV.Prologue.fiber_completeness_count :canonical_fiber_completeness.fiber_sectors.length = 3

[V.R04] Fiber completeness: exactly 3 fiber sectors exist on T².

Tau.BookV.Prologue.base_completeness_count

source theorem Tau.BookV.Prologue.base_completeness_count :canonical_fiber_completeness.base_sectors.length = 2

Base completeness: exactly 2 base sectors exist on τ¹.

Tau.BookV.Prologue.fiber_covers_nongrav

source theorem Tau.BookV.Prologue.fiber_covers_nongrav :BookIV.Sectors.em_sector.generator = Kernel.Generator.gamma ∧ BookIV.Sectors.strong_sector.generator = Kernel.Generator.eta ∧ BookIV.Sectors.higgs_sector.generator = Kernel.Generator.omega ∧ BookIV.Sectors.em_sector.depth ≥ 2 ∧ BookIV.Sectors.strong_sector.depth ≥ 2 ∧ BookIV.Sectors.higgs_sector.depth ≥ 2

[V.T06] The boundary holonomy algebra restricted to fiber sectors B, C, Omega covers all non-gravitational physics.

Fiber sectors provide:

  • B (EM): photon transport, Maxwell equations, fine structure

  • C (Strong): color holonomy, confinement, mass gap

  • Omega (Higgs): mass generation, chirality crossing

The fiber carrier type assignment agrees with sector physics.

Tau.BookV.Prologue.temporal_complement_recap

source theorem Tau.BookV.Prologue.temporal_complement_recap :BookIV.Sectors.kappa_AA.numer + BookIV.Sectors.kappa_DD.numer = BookIV.Sectors.kappa_AA.denom

[V.R05] Temporal complement recap: κ(A;1) + κ(D;1) = 1 from Book IV.

This identity means the base sectors (Gravity + Weak) fully account for the temporal coupling budget. The temporal pair is hermetically closed: no coupling leaks between temporal and spatial sectors.

Wraps Tau.BookIV.Arena.temporal_complement.

Tau.BookV.Prologue.hermetic_base_fiber

source theorem Tau.BookV.Prologue.hermetic_base_fiber :canonical_fiber_completeness.fiber_sectors.length + canonical_fiber_completeness.base_sectors.length = 5

The hermetic principle: base (2) + fiber (3) = 5 total sectors. This partition is exact — every sector lives on exactly one carrier.

Tau.BookV.Prologue.holonomy_covers_all

source theorem Tau.BookV.Prologue.holonomy_covers_all :BookIV.Arena.holonomy_generators.length = 5

The holonomy generators cover all 5 sectors (from Book IV).